Pluralism and Normativity in Logic

Originally written for PHIL 2710 Recent Debates in the Philosophy of Logic with Prof. Joshua Schechter.

Logical pluralism is the view that there is more than one “correct” logic. On this view, there is no single content of logical concepts like consequence and validity, rather such concepts are multipartite, made up of different logics that are appropriate for different uses. Here, I will focus on the treatment of pluralism given in Beall and Restall’s 2000 article “Logical Pluralism” and its criticisms in Griffiths and Paseau (2022). First, I will defend Beall and Restall’s so-called “modest pluralism” from the motivation objection given by Griffiths and Paseau, which will then lead me to a deeper evaluation of Beall and Restall’s project. Ultimately, I will conclude that Beall and Restall insufficiently motivated their pluralism, but that such a motivation does exist, derived from a modern acceptance of the looseness of the normative connection between logic and reasoning. Looking clearly through this lens at logic, I argue that Beall and Restall’s modest pluralism is natural and promising.

An outline of Beall and Restall’s position is as follows: the essence of logic is logical consequence, and the essence of logical consequence is the dogma that “a conclusion A follows from premises 𝝨 if and only if any case in which each premise in 𝝨 is true is also a case in which A is true” (Beall and Restall 2000 476). This principle, which they call (V), is by their lights the core of the pretheoretic notion of logical consequence, and any elaboration of what a “case” is therefore constitutes a legitimate logic. They do little in the way of defending the claim that (V) is the “heart of logical consequence” (Beall and Restall 2000 477). Seemingly, this characterization is plainly obvious to them, with its only explicit backing coming from a vague gesture at its widespread use in logic literature. Beall and Restall go on to give some examples of elaborations of “cases,” such as Tarskian models which lead to classical logic, situations which lead to relevant logic, and constructions which lead to intuitionistic logic.

Their first example, however, is perhaps the most interesting for an analysis of their view. If “cases” are taken simply to be possible worlds, then (V) gives a “necessary truth preservation” account of consequence. This account, by their own admission, “is not at all the traditional picture of logical consequence” (Beall and Restall 2000 478). This is because it is formal or topic neutral. Using their example, it turns out that on this account “the ball is colored” is a logical consequence of “the ball is red” since, in any possible world where the ball is red, it is also colored. Note, though, that if one were to formalize such an argument it would look like a move from “X is P” to “X is Q” which is, of course, not a valid form in general. The validity of the argument on the necessary truth preservation account depends on the actual content of the properties of being red and being colored. Thus, logical consequence here is neither formal (i.e. a pure feature of the forms of the propositions involved) nor is it topic neutral (i.e. unreliant on the content of the concepts involved.) Some see this generality, formality, and topic-neutrality of logic as a core feature (for example, see Sher (1996).) Beall and Restall, on the other hand, are unfazed by the discovery that one way of filling in the notion of “cases”in (V) leads to a loss of formality, and do not even consider whether that fact bears at all on whether their elaboration of the pretheoretic notion was appropriate. Clearly, the foundational notion of logical consequence on which this pluralism is based (via the guiding (V) principle) must be interrogated more thoroughly. I will return to this idea soon.

Concerns such as the one just presented about formality are meant to be ironed out in the treatment Beall and Restall gave of the view in their 2006 book (also called Logical Pluralism.) The new view abandons the idea that (V) is the only criterion for a logic (now its version of consequence must also be “necessary, normative, and formal” (Beall and Restall 2006 35) for example.) I will argue that this updated view is weaker than the original, and my primary consideration will be of the original 2000 view.

To round out the summary of Beall and Restall’s view, it is worth pointing out that many structures that carry the mantle of “logics” are excluded from the term here. Only a select few from the vast catalog of systems of inference rules successfully satisfy the (V) criterion. This is not a necessary feature of logical pluralism. By comparison, the pluralism presented in Shapiro (2014) is significantly more lax, essentially endorsing any nontrivial, mathematically interesting system.

This contrast is, according to Griffiths and Paseau, a fatal flaw for Beall and Restall’s view. Dubbing the view “modest pluralism” as compared to the far more permissive “eclectic pluralism” of Shapiro, they contend that the choosiness of the view generates an arbitrariness concern. That is, they think that Beall and Restall are unjustified in their exclusion of logics that fail to be an elaboration of (V), and that modest pluralism therefore effectively collapses into eclectic pluralism. This is important for their larger argument against pluralism, which hinges on the difficulty of finding an appropriate metalogic for a pluralist system. In other words, if multiple logics are correct, which one should we use when reasoning about logic itself? It seems that the very same object-level pluralist arguments could be deployed against any claim that there is a single correct logic for the role. Some alternatives might be to pick a logic that consists of the intersection of all the admissible logics (or is in some other way at least as weak as all of them) or to pick a candidate logic arbitrarily, using it without claiming it is the sole correct choice to use. Strategies like this pose serious problems for eclectic pluralism, which has a menagerie of logics so vast that, as Griffiths and Paseau argue, its intersection will be nearly empty and many of its candidate logics will be far too weak to suitably serve as the metalogic.

Note, however, that modest pluralism does not suffer from this problem. Since the roster of admissible logics in modest pluralism is relatively sparse, a strategy to the metalogic involving intersections or the like becomes available once again. Thus, the overall dismissal of pluralism from Griffiths and Paseau hinges on eclectic pluralism being the only motivated form of pluralism.

Let us now examine, then, why Griffiths and Paseau believe that modest pluralism is untenable and collapses into eclectic pluralism. Their work mainly draws on the 2006 presentation of Beall and Restall’s ideas, in which they justify their principle (V) (which they there call the Generalized Tarski Thesis) not by an appeal to the pretheoretic notion of consequence per se, but instead to its “settled core.” Griffiths and Paseau, in a reasonable move, interpret the notion of the settled core as a sociological one. In order to determine whether some feature holds of the settled core of logical consequence, one should look to the ongoing debate in the logical literature and see whether it is an active point of contention. Only if there is widespread agreement that logic must have the feature in question is it a part of the settled core of the concept. Rightly, Griffiths and Paseau point out that with this empirical test almost nothing is part of the settled core. In the logical literature, as in the rest of academic literature, debate rages on about almost every issue. They point out debate over features that Beall and Restall, in their 2006 work, do take to be part of the settled core such as objections to formality and necessity, and point to features such as “suitability for modeling mathematical discourse” and axiomatizability that are accepted as frequently but Beall and Restall do not take as part of the settled core. Beall and Restall’s elaboration of the settled core is unmotivated since the features they include are not particularly more settled in the literature than the ones they exclude. Thus, since this settled core was the criterion by which they selected which logics were admissible and which were not, their entire pluralism is unmotivated, and thereby collapses into a more wide-ranging eclectic pluralism.

This argument is a damning one, but only to the “settled core” methodology and not to the overall project of modest pluralism. Looking only towards consensus among contemporary philosophers is almost never a recipe for capturing the truly essential features of a concept, or a good pretheoretic intuition of the concept, or really anything at all because of the truly wide and deep nature of philosophical debate. It cannot be that the way to choose which logics are good and which are bad is by looking to those which everyone will agree to, since there will always be an uncooperative few who always disagree. Notably, the settled core idea does not appear in the 2000 paper, and the state of debate in the literature only takes a central role for Beall and Restall in their 2006 book. If we reject the use of the settled core, we might still be able to salvage modest pluralism by refocusing on Beall and Restall’s 2000 account.

The flipside of rejecting the settled core, however, is that one cannot look to academic disagreement to motivate pluralism. There is a rough implicit argument in the 2000 presentation of Beall and Restall’s view that might be explicated as follows: There are many different logics, each of which is internally coherent and has its supporters. The different factions are not talking past each other, and each of their preferred logics is a viable candidate for the “one true logic.” Thus, there is no one true logic after all and each of the candidates is right in its own way.

Shapiro, in his presentation of his own view, refers to pluralism as being akin to a “folk-relativism” about logic (Shapiro 63). I think the argument form sketched above comes dangerously close to one that, if it were to be valid in general, would easily endorse folk-relativism about a wide range of philosophical concepts in an unappealing way. For example, utilitarianism and deontology are competing elaborations of ethics. Each has its supporters and is internally coherent. The factions do not seem to be talking past one another, but rather seem to be presenting serious alternative contenders for a genuine elaboration of the shared core concept of ethics. Can we conclude from this situation that we should be pluralists about ethics? Surely, this argument would be unconvincing to almost anyone that was not already partial to a form of moral relativism. A similar line of thought could be used to justify “pluralist” (that is, anti-realist or relativist) conclusions about all sorts of other hotly contested philosophical concepts like causation, knowledge, mathematics, and so forth. It is not in general valid to observe contention between seemingly viable candidates and from this observation alone derive pluralism. This, however, is dangerously close to what Beall and Restall do in their 2000 work. They present the relatively uncontroversial principle (V) as being a base characterization of logic. Then, they present multiple ways to fill in the principle’s details, each of which results in a coherent theory with supporters. From this, they conclude that the seemingly rival theories all actually have coequal status.

In their criticism of the settled core, Griffiths and Paseau point to a very interesting feature of logic that, they claim, is as generally held as central as the features selected by Beall and Restall: monism. We have already seen why disputes about the settled core as regards the state of debate in the literature is not a good way to conduct an investigation of logic, but this still presents a crucial question: why is it not simply baked into the idea of logic that there is a single correct choice? Pluralism is a somewhat new and obscure theory, so if an analysis of logic begins with a pretheoretic notion why is it that the notion can admit pluralism at all? This is a serious concern, and one that motivates a return to careful consideration of the “pretheoretic notion” and methodology more generally.

Broadly speaking, a very common philosophical methodology runs along the lines of beginning with a pretheoretic, intuitive notion of a concept, then crafting a theory that models the pretheoretic notion. Very frequently, one then finds that the natural choice for such a theory does not in fact satisfy all of the desiderata imposed by the pretheoretic notion, and one is forced either to reconsider prior intuitions or to engage in further theorycrafting, and frequently both. In this dialectical manner, the target notion is refined from the original pretheoretic intuitive one at the same time that the theory is developed. For example, upon Edmund Gettier’s discovery that a “justified true belief” account includes cases that we intuitively do not think to be knowledge, philosophers were forced to take a critical look at such intuitions while simultaneously building new candidate analyses for the concept of knowledge. If a conflict arises between intuitions and theory, this procedure only directs to revise at least one so as to eliminate it (it is available either to endorse Gettier cases as knowledge, or to build a new theory that excludes them, or something in between.) Sometimes, we find that no theory can be everything that our intuitions initially want the concept to be. When there are multiple candidate theories, it is because the complete spectrum of desiderata generated by refined intuition is impossible to fully satisfy simultaneously, and so one theory will better cohere with one aspect of intuition and another will allow us to keep a different intuition.

Laying the general philosophical methodology like that may seem facile, and indeed this framework is so broad as to be a good description of concept analysis in subfields ranging from ethics to metaphysics to epistemology. However, returning to logical consequence with this procedure firmly in mind will rectify the errors that we have encountered thus far. Two features of the process are salient when it comes to the discussion we have already seen.

First, this process makes no direct reference to the state of discourse about the concept, or to any sociological academic phenomena at all. The shared use of the concept plays a part in selecting appropriate intuitive starting ideas, but at no point does this procedure license the use of a settled core or any similar idea wherein uncontroversial elements of the concept are given priority.

Second, the procedure gives us a good way to tell when we have arrived at a justified relativism or pluralism. If we have revised our target concept in such a way that there are multiple theories that fulfill all of the desiderata, then that is good reason to think that all such theories are coequally correct. This is because our methodology for selecting between theories was to compare their different successes and failings in cohering with our target concept. If there are no more pros and cons, if we have multiple theories that give us everything we want for the concept, then we have a justified pluralism between those theories.

Thus, a huge amount hinges on both the pretheoretic notion and its successful refinement. Beall and Restall do not fully justify their principle (V) on these grounds, but that is not a reason to think that it fails to be the final target concept for logical consequence. What their work does show is that if principle (V) is essentially what we expect of logic, no more and no less, then we should be pluralists. Let us, then, apply the above procedure to the concept of logical consequence.

One aspect of logic that should be included in our very starting point of intuition is a tight-knit connection with reasoning. Logic is often presented as being the stuff of reasoning, and it is at first quite difficult to think about what it might be to intentionally abandon logic when reasoning. Thus, among the various other aspects, a first pass at intuitive desiderata for logical consequence should include a tight normative restriction for thought. However, as demonstrated in Harman (1986), this leads to trouble quickly. If we want logical consequence to have the property that B being a consequence of A means that one should believe B when one believe A, we find that one is licensed to believe a ridiculous claim only because it is the consequence of a ridiculous claim one already believes. Likewise, various other problems make it so that the normative aspect of logic cannot be particularly tight. MacFarlane (2004) investigates various possible “bridge principles” connecting logical consequence to thought, but finds that there is no one great choice, and that the most plausible options are relatively weak. In other words, then, the result of the dialectic between intuition and theory when it comes to normativity has led us to abandon the hope of having a logic with an extremely tight connection to reasoning. We should not expect logic to be “constitutive” of reasoning, or that it imposes significantly stricter, or even different, norms on our reasoning than other epistemic obligations. In our search for an appropriate concept of logical consequence and a satisfactory theory for that concept, our methodological procedure has led us to significantly tone down our normative expectations.

After this step of desiderata-refinement, I will argue that many other preliminary intuitions about logic are shorn away. In other words, several intuitive ideas about logic come from the impression that logic should be tightly normative for thought, which we can now see is an error. One reason for this is that the normative supremacy of logic for thought is what generates a lot of the intuitive “specialness” we assign to logic. Once it turns out that logic and thought are somewhat more distant relations, logic ceases to feel so special, and many of our intuitive desires for it turn out to be unjustified.

For example, as discussed above there may be an initial inclination towards monism as a core property. In other words, the concept of logical consequence should be fundamentally united and single. I think the primary reason for such an intuition is the intuitive desire for normativity. If it were the case that logic were constitutive or otherwise extremely tightly binding for thought, then we would indeed want a united concept, since it would be schizophrenic, if not outright contradictory, to have multiple consequence relations simultaneously govern one’s thought. With a newfound clearheaded view of normativity, one is hard-pressed to find an intuitive reason that pluralism is impossible. This is not to say, of course, that this decides the matter in favor of pluralism. There may be theoretical considerations that disqualify pluralism (for example, concerns like Griffiths and Paseau’s about metalogic) and in order to vindicate pluralism one has to actually demonstrate multiple logics that all satisfy all our desires for logic. However, it eliminates the worry that one of our conceptual goals for logic is monism itself. Pluralism can, so to speak, get off the ground.

Another feature which starts to have shakier footing in light of reduced normativity is formality. One serious reason to not want logic to incorporate particular contents is that knowledge of particular things might not be accessible a priori. This would be a serious problem if logic were tightly normative, since it would generate many completely infeasible obligations for thought. With a looser normativity, however, a type of consequence like necessary truth preservation seems less dangerous, since by calling it an admissible logic we are no longer committing ourselves to being normatively bound by it. Indeed, a full acknowledgement of the “unspecialness” of logic should make it harder to separate out the notion of logical consequence from that of consequence in general. This should also make us more generally amenable to pluralism about logic, since the principal, guiding purpose of logic for use in thought has dimmed considerably. A view that sees multiple situations and purposes for logic as coequal should seem significantly more reasonable.

With these considerations in mind, it seems to me reasonably justified to say that Beall and Restall’s (V) formulation is, in fact, a good terminus for the target concept of logical consequence. Its truth-preservation formula can hardly be disputed as a model of consequence in general, and its lack of other distinguishing logical aspects can be vindicated in light of the normative unspecialness of logic.

Therefore, I think that if one takes seriously the light normativity of logic, then modest pluralism is a natural result. A challenge to such a view would consist in putting forward more desiderata for logic that are not rooted in holding on too tightly to normativity. As I have shown, I think almost all of the classic characterizations of logic other than those capture in (V) do have normativity as their base, but such a challenge is possible.

Works Cited

Beall, JC, and Greg Restall (2000). “Logical Pluralism.” Australasian Journal of Philosophy 78(4): 475–493. https://doi.org/10.1080/00048400012349751

Beall, JC, and Greg Restall (2006). Logical Pluralism. Oxford: Oxford University Press.

Griffiths, Owen, and A. C. Paseau (2022). One True Logic: A Monist Manifesto. Oxford: Oxford University Press. https://doi.org/10.1093/oso/9780198829713.001.0001

Harman, Gilbert (1986). Change in View: Principles of Reasoning. Cambridge, MA: MIT Press.

MacFarlane, John (2004). “In What Sense (If Any) Is Logic Normative for Thought?”

Shapiro, Stewart (2014). Varieties of Logic. Oxford: Oxford University Press. https://doi.org/10.1093/acprof:oso/9780199696529.001.0001

Sher, G. Y. (1996). “Did Tarski Commit ‘Tarski’s Fallacy’?” The Journal of Symbolic Logic 61(2): 653–686. https://doi.org/10.2307/2275681